Can a rule be formulated to explain this to 7 year old? I'm trying to teach math to my 7 year old daughter. I'm teaching following type of equations. 
$$\cdots  - x = y$$ 
I'm able to explain her the rule that: 

when $\cdots- x = y$, we can always take $x$ (value on the left of equation) to the other side of $=$ sign and flip( $-$ to $+$ and vice versa), $-$ to $+$ and get the answer. 

Meaning when $\cdots - x = y$, we can always do $\cdots =  y + x$ and get the answer. 
This rule works for 
\begin{align*}
x + \cdots &= y \\
\cdots + x &= y\\ 
\cdots - x &= y \\
\end{align*}
But it doesn't work for $x - \cdots = y$. Because if you apply the rule, you get - (answer) and not just ( answer ) 
My question is given that I'm trying to teach this to 7 year old, is there any better method where one rule would cover all 4 cases? Any ideas, thoughts...
\begin{align*}
 - x + \cdots &= y\\
 -\cdots + x &= y\\
 - \cdots  - x &= y\\
 - x  - \cdots &= y
\end{align*}
 A: You shouldn't be using "rules" at all; that is not what mathematics is about, at any level.  (This is an enormous pet peeve of mine.  There is a commercial floating around Hulu about some kind of online tutoring program where a woman describes to a girl the rule for computing the area of a triangle given its base and height, and then completely fails to draw the diagram that explains why this rule works.  It annoys me to no end.  (This particular example is also brought up in the infamous Lockhart's lament.))  
I have some amount of money in my bank account.  When I withdraw $x$ dollars, I have $y$ dollars left.  How much money did I have originally?  $x + y$.  How much money do I have now?  $(x + y) - x = y$.  
I have some amount of money in my bank account.  When I deposit $x$ dollars, I now have $y$ dollars.  How much money did I have originally?  $y - x$.  How much money do I have now?  $(y - x) + x = y$.
At some point it is probably a good idea to mention that $x + y$ is the same as $y + x$ (that is, depositing $x$ dollars and then depositing $y$ dollars is the same as depositing $y$ dollars and then depositing $x$ dollars).  Then you've covered all of the "cases."
Alternately, a physical analogy ought to work well.  I am some distance away from a wall.  When I move $x$ feet towards the wall, I am $y$ feet away from the wall.  How far was I originally away from the wall?  $x + y$.  How far am I away from the wall now?  $(x + y) - x = y$.
I am some distance away from a wall.  When I move $x$ feet away from the wall, I am $y$ feet away from the wall.  How far was I originally away from the wall?  $y - x$.  How far am I away from the wall now?  $(y - x) + x = y$.
A: I'm not clear on what the "rule" you say "doesn't work" is... Still...
As Qiaochu says, don't do "rules". The key to all of these manipulations is:

If two things are equal, and you do the same thing to each of them, the results will also be equal.

So, if $A$ is equal to $B$, then adding $2$ to $A$ will result in the same thing as adding $2$ to $B$: if $A=B$, then $A+2 = B+2$. 
If you have $\cdots - x = y$, then you have two things that are equal. Adding $x$ to both will still give you equal things, so
$$(\cdots - x) + x = y + x.$$
Then using the fact that $-x+x = 0$, you get $\cdots = y+x$. 
All of the manipulations you propose are instances of this: if you have two equal things, and you do the same thing to both, the results are still equal. 
A: "Do the same thing to both sides" also works for other operators - multiply, divide, powers, roots ... you could say it's the master/meta rule of all the others.
